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normal of plane

The perpendicular or normal line of a plane is a special case of the surface normal, but may be defined separately as follows:

A line lll is a normal of a planeππ\pi, if it intersects the plane and is perpendicular to all lines passing through the intersection point in the plane. Then the plane ππ\pi is a normal plane of the line lll. The normal plane passing through the midpoint of a line segment is the center normal plane of the segment.

There is the

Theorem. If a line (lll) cuts a plane (ππ\pi) and is perpendicular to two distinct lines (mmm and nnn) passing through the cutting point (LLL) in the plane, then the line is a normal of the plane.

..lllππ\piLLLMMMNNNAAAPPPQQQmmmnnnaaa

Proof. Let aaa be an arbitrary line passing through the point LLL in the plane ππ\pi. We need to show that a⟂lperpendicular-toala\perp l. Set another line of the plane cutting the lines mmm, nnn and aaa at the points MMM, NNN and AAA, respectively. Separate from lll the equally long line segments L⁢PLPLP and L⁢QLQLQ. Then

Thus the segments P⁢APAPA and Q⁢AQAQA, being corresponding parts of two congruent triangles, are equally long. I.e., the point AAA is equidistant from the end points of the segment P⁢QPQPQ, and it must be on the perpendicular bisector of P⁢QPQPQ. Therefore A⁢L⊥P⁢QbottomALPQAL\bot PQ, i.e. a⊥lbottomala\bot l.

Proposition 1. All normals of a plane are parallel. If a line is parallel to a normal of a plane, then it is a normal of the plane, too.

Proposition 2. All normal planes of a line are parallel. If a plane is parallel to a normal plane of a line, then also it is a normal plane of the line.